The phrase 'When in Rome, do as the Romans do' has no specific original story. It's more of a common saying that emerged over time to suggest adapting to local customs and behaviors when in a different place.
It's a form of family bonding. Telling stories is a shared activity that can bring the family closer together. The sons can feel a connection with their fathers and the family's past through these stories.
Sure. Since Percy Jackson books are quite popular around the world, there's no reason to think that people in Rome wouldn't read the fanfiction. The books have made Greek and Roman mythology exciting for a new generation. Fanfiction allows fans to further explore the characters and world, and with Rome having a connection to the mythology on which the series is based, it's entirely possible that Romans would be among the readers of Percy Jackson fanfiction.
They might tell stories about great Roman heroes like Julius Caesar, his military conquests and how he shaped Rome. These stories are to inspire their sons to be brave and ambitious.
They might read it for entertainment. The Last Olympian has an exciting world full of magic and adventure, and fanfiction allows Romans to explore different aspects of that world that the original might not have covered fully.
Maybe they are interested in the story's unique setting and plot twists. The 'Battle of the Labyrinth' likely has elements that appeal to their sense of adventure and mystery.
One reason could be the allure of the crossover aspect. Since 'The Last Olympian' is based on Greek mythology and the Romans had their own take on mythology which was related. Reading fanfiction allows them to see how the two worlds could interact in ways not shown in the original works.